Thursday, August 13, 2015

"Wessel's fame as a mathematician rests solely
on this paper, which was published in 1799,
giving for the first time a geometrical interpretation
of complex numbers. Today we call this geometric
interpretation the Argand diagram but Wessel's
work came first. It was rediscovered by Argand
in 1806 and again by Gauss in 1831. ….

Of course it is not unreasonable to call the
geometrical interpretation of complex numbers
the Argand diagram since it was Argand's work
which was influential. It was so named before
the world of mathematics learnt of Wessel's prior
publication. In fact Wessel's paper was not
noticed by the mathematical community until 1895…."

Monday, October 13, 2014

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric or, less formally, geomagic is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles or pieces ) may appear in the cells instead of numbers.”

Saturday, January 7, 2012

" … debates about realism, that is, those debates that ask
whether or not one or another aspect of the world is independent
of the way we represent that aspect to ourselves. For example,
is there a realm of mathematical entities that exists fully formed
independently of our mathematical activity? Are there facts about
the past that our use of the past tense aims to capture?"

In memory of time I spent on Wilcox Avenue
in the Hollywood of the early sixties, seeHans Reichenbach on the use of the past tense
(section 51 of Elements of Symbolic Logic,
New York: The Macmillan Company, 1947, 287–98.)

The day begins with Yvonne’s arrival at the Bella Vista bar in Quauhnahuac. From outside she hears Geoffrey’s familiar voice shouting a drunken lecture this time on the topic of the rule of the Mexican railway that requires that “A corpse will be transported by express!” (Lowry, Volcano, p. 43).

Kylie

Finney

Well if you want to ride you gotta ride it like you find it.Get your ticket at the station of the Rock Island Line.

“Wessel’s paper [in Danish] was not noticed by the mathematical community until 1895… A French translation… was published in 1897 but an English translation of this most remarkable work was not published until 1999 (exactly 200 years after it was first published)….

We have called Wessel’s work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel’s contribution was [1]:-

… superior to and more modern in spirit to Argand’s.

In the [1] article the approaches by Argand and Wessel are compared and contrasted. Of course Wessel was a surveyor and his paper was motivated by his surveying and cartography work:-

Wessel’s development proceeded rather directly from geometric problems, through geometric-intuitive reasoning, to an algebraic formula. Argand began with algebraic quantities and sought a geometric representation for them. … Wessel’s initial formulation was remarkably clear, direct, concise and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited.

However more is claimed for Wessel’s single mathematical paper than the first geometric interpretation of complex numbers. In [3] Crowe credits Wessel with being the first person to add vectors. Again this shows the depth of Wessel’s thinking but again, as the paper was unnoticed it had no influence on mathematical development despite appearing in the Memoirs of the Royal Danish Academy which by any standard was a major source of publications….

Wednesday, February 20, 2008

"Oppenheimer had learned Sanskrit at Berkeley so as to read the Gita in the original; he always kept a worn pink copy on the bookshelf closest to his desk. It is therefore likely that he may have actually thought of the original, Sanskrit, verse rather than the English translation. The closest that fits this meaning is in the 32nd verse from the 11th chapter of the Gita.

kalosmi lokaksaya krt pravrddho

This literally means: I am kAla, the great destroyer of Worlds. What is intriguing about this verse, then, is the interpretation of kAla by Jungk and others to mean death. While death is technically one of the meanings of kAla, a more common one is time."

"KAla" (in the Harvard-Kyoto transliteration scheme) is more familiar to the West in the related form of Kali, a goddess sometimes depicted as a dancing girl; Kali is related to kAla, time, according to one website, as "the force which governs and stops time." See also the novel The Fermata, by Nicholson Baker.

The fact that Oppenheimer thought of Chapter 11, verse 32, of the Gita may, as a mnemonic device, be associated with the use of the number 1132 in Finnegans Wake.

Thursday, June 14, 2007

"Kurt Waldheim, the former United Nations Secretary General and President of Austria whose hidden ties to Nazi organizations and war crimes was [sic] exposed late in his career, died today at his home in Vienna. He was 88." —The New York Times this afternoon

Related material:

From a story by
Leonard Michaels
linked to on
Aaron Sorkin's
birthday, June 9:

"Induction and analogy, in which he was highly gifted, were critical to mathematical intelligence.

It has been said that the unexamined life isn't worth living. Nachman wasn't against examining his life, but then what was a life? ….

… As for 'a life,' it was what you read about in newspaper obituaries. He didn't need one. He would return to California and think only about mathematics."

Thursday, June 7, 2007

His early novels, written in two weeks or less, were published in double-decker Ace paperbacks that included two books in one, with a lurid cover for each. “If the Holy Bible was printed as an Ace Double,” an editor once remarked, “it would be cut down to two 20,000-word halves with the Old Testament retitled as ‘Master of Chaos’ and the New Testament as ‘The Thing With Three Souls.'”

“… In this way, mystical as well as psychological dimensions are established. Later on, the two pass by a printer’s shop window and curiously stop to inspect, amidst wedding portraits and well in front of the revolving flywheel of the printing machines, ‘a photographic enlargement purporting to show the disintegration of a glacial deposit in the Sierra Madre, of a great rock split by forest fires.’ Significantly the picture is called ‘La Despedida,’ the Parting. Yvonne cannot help but see the symbolic significance of the photograph and wishes with all of her might ‘to heal the cleft rock’ just as she wishes to heal the divorce….”

Some method in this madness is revealed by the evening lottery number, 814, which leads to an entry of 8/14:

“Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods….”

"This literally means: I am kala, the great destroyer of Worlds. What is intriguing about this verse, then, is the interpretation of kala by Jungk and others to mean death. While death is technically one of the meanings of kala, a more common one is time."

Frank Kelly Freas, who created, among other works, 400 portraits of saints for the Franciscans and the covers of Mad Magazine from 1958 through 1962. "I found it difficult to shift my artistic gears from the sublime to the ridiculous and back again," he said of his departure from Mad.

Will Eisner, "an innovative comic-book artist who created the Spirit, a hero without superpowers, and the first modern graphic novel."

Yesterday's entry provided an approach to The Dark Lady, Kali, that was, in Freas's apt word, "ridiculous." The illustration below, "Mate," is an attempt to balance yesterday's entry with an approach that is, if not sublime, at least more serious. It is based on a similar illustration from Jan. 31, 2003, with actress Judy Davis playing The Dark Lady. Today it seems appropriate to replace Davis with another actress (anonymous here, though some may recognize her). I once knew her (unlike Davis) personally. One of my fondest memories of high school is reading Mad Magazine with her in the school lunch room. Our lives diverged after high school, but I could happily have spent my life in her company.

He is sometimes called the "father of the A-bomb." He said that at the time of the first nuclear test he thought of a line from the Sanskrit holy book, the Bhagavad Gita: "I am become death, the destroyer of worlds." The following gives more details.

The Bomb of the Blue God

M. V. Ramana

Center for Energy and Environmental Studies, Princeton University

Published in SAMAR: South Asian Magazine for Action and Reflection, Issue 13

Oppenheimer had learned Sanskrit at Berkeley so as to read the Gita in the original; he always kept a worn pink copy on the bookshelf closest to his desk. It is therefore likely that he may have actually thought of the original, Sanskrit, verse rather than the English translation. The closest that fits this meaning is in the 32nd verse from the 11th chapter of the Gita.

kalosmi lokaksaya krt pravrddho

This literally means: I am kAla, the great destroyer of Worlds. What is intriguing about this verse, then, is the interpretation of kAla by Jungk and others to mean death. While death is technically one of the meanings of kAla, a more common one is time. Indeed, the translations of the Gita by S. Radhakrishnan, A. C. Bhaktivedanta, Nataraja Guru and Eliot Deutsch say precisely that. One exception to this, however, is the 1929 translation by Arthur Ryder. And, indeed, in a 1933 letter to his brother, Robert Oppenheimer does mention that he has "been reading the Bhagavad Gita with Ryder and two other Sanskritists." The misinterpretation, therefore, may not have been the fault of Oppenheimer or Jungk. Nevertheless, the verse does not have anything to do with an apocalyptic or catastrophic destruction, as most people have interpreted it in connection with nuclear weapons. When kAla is understood as time, the meaning is drastically changed to being a reminder of our mortality and finite lifetimes ­ as also the lifetimes of everything else in this world (including plutonium and uranium, despite their long, long, half-lives!). It then becomes more akin to western notions of the "slow march of time" and thus having little to do with the immense destruction caused by a nuclear explosion. While the very first images that arose in the father of the atomic bomb are a somewhat wrong application of Hindu mythology, his recollection of the Bhagavad Gita may have been quite pertinent. As is well known, the Bhagavad Gita was supposedly intended to persuade Arjuna to participate in the Kurukshetra battle that resulted in the killing of thousands. Thus, Oppenheimer may well have been trying to rationalize his involvement in the development of a terrible weapon.

"KAla" (in the Harvard-Kyoto transliteration scheme) is more familiar to the West in the related form of Kali, a goddess sometimes depicted as a dancing girl; Kali is related to kAla, time, according to one website, as "the force which governs and stops time." See also the novel The Fermata, by Nicholson Baker.

The fact that Oppenheimer thought of Chapter 11, verse 32, of the Gita may, as a mnemonic device, be associated with the use of the number 1132 in Finnegans Wake.

Chorus:
Cheeseburger in paradise (paradise)
Makin' the best of every virtue and vice (paradise)
Worth every damn bit of sacrifice (paradise)
To get a cheeseburger in paradise
To be a cheeseburger in paradise
I'm just a cheeseburger in paradise!"

For some, paradise — or at least the gateway to paradise — is at Puerto Vallarta, Mexico.

"Cheeseburger In Paradise, Puerto Vallarta, opened for business on November 7, 1999." — The same date, mentioned in last night's "Whirligig" entry, that Fox Studios Australia opened in Sydney with a song by Kylie Minogue.

Tonight is twelfth night in Australia; 4 AM Jan. 5
in New York City is 8 PM Jan. 5 in Sydney.

An October 6 entry:

Twenty-first Century Fox

On Sunday, October 6, 1889, the Moulin Rouge music hall opened in Paris, an event that to some extent foreshadowed the opening of Fox Studios Australia in Sydney on November 7, 1999. The Fox ceremonies included, notably, Kylie Minogue singing "Diamonds are a Girl's Best Friend."

Red Windmill

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

In accordance with the twelfth-night
"whirligig of time" theme,
here are two enigmatic quilt blocks:

Devil's Claws, or
Hourglass Var. 3

Yankee Puzzle, or
Hourglass Var. 5

One can approach these symbols in either a literary or a mathematical fashion. For a purely mathematical discussion of the differences in the two symbols' structure, see Diamond Theory. Those who prefer literary discussions may make up their own stories.

"Plato is wary of all forms of rapture other than reason's. He is most deeply leery of, because himself so susceptible to, the literary imagination. He speaks of it as a kind of holy madness or intoxication and goes on to link it to Eros, another derangement that joins us, but very dangerously, with the gods."